A phase-field model for liquid–gas mixtures: mathematical modelling and discontinuous Galerkin discretization

To model a liquid–gas mixture, in this article we propose a phase-field approach that might also provide a description of the expansion stage of a metal foam inside a hollow mold. We conceive the mixture as a two-phase incompressible–compressible fluid governed by a Navier–Stokes–Cahn–Hilliard system of equations, and we adapt the Lowengrub–Truskinowsky model to take into account the expansion of the gaseous phase. The resulting system of equations is characterized by a velocity field that fails to be divergence-free, by a logarithmic term for the pressure that enters in the Gibbs free-energy expression and by the viscosity that degenerates in the gas phase. In the second part of the article we propose an energy-based numerical scheme that, at the discrete level, preserves the mass conservation property and the energy dissipation law of the original system. We use a discontinuous Galerkin approximation for the spatial approximation and a modified midpoint based scheme for the time approximation.